Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 5.4s

Alternatives: 6

Speedup: 0.6×

• 24.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot y\right) \cdot \left(1 - y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (* x y) (- 1.0 y)))

C

double code(double x, double y) {
	return (x * y) * (1.0 - y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) * (1.0d0 - y)
end function

Java

public static double code(double x, double y) {
	return (x * y) * (1.0 - y);
}

Python

def code(x, y):
	return (x * y) * (1.0 - y)

Julia

function code(x, y)
	return Float64(Float64(x * y) * Float64(1.0 - y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) * (1.0 - y);
end

Wolfram

code[x_, y_] := N[(N[(x * y), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot y\right) \cdot \left(1 - y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (* x y) (- 1.0 y)))

C

double code(double x, double y) {
	return (x * y) * (1.0 - y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) * (1.0d0 - y)
end function

Java

public static double code(double x, double y) {
	return (x * y) * (1.0 - y);
}

Python

def code(x, y):
	return (x * y) * (1.0 - y)

Julia

function code(x, y)
	return Float64(Float64(x * y) * Float64(1.0 - y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) * (1.0 - y);
end

Wolfram

code[x_, y_] := N[(N[(x * y), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(\left(1 - y\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* y (* (- 1.0 y) x)))

C

double code(double x, double y) {
	return y * ((1.0 - y) * x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y * ((1.0d0 - y) * x)
end function

Java

public static double code(double x, double y) {
	return y * ((1.0 - y) * x);
}

Python

def code(x, y):
	return y * ((1.0 - y) * x)

Julia

function code(x, y)
	return Float64(y * Float64(Float64(1.0 - y) * x))
end

MATLAB

function tmp = code(x, y)
	tmp = y * ((1.0 - y) * x);
end

Wolfram

code[x_, y_] := N[(y * N[(N[(1.0 - y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]

2. Taylor expanded in x around 0 99.9%

   \[\leadsto \color{blue}{y \cdot \left(\left(1 - y\right) \cdot x\right)} \]

3. Final simplification 99.9%

   \[\leadsto y \cdot \left(\left(1 - y\right) \cdot x\right) \]

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+43} \lor \neg \left(y \leq 10^{+16}\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.8e+43) (not (<= y 1e+16)))
   (* y (* y (- x)))
   (* x (- y (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7.8e+43) || !(y <= 1e+16)) {
		tmp = y * (y * -x);
	} else {
		tmp = x * (y - (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-7.8d+43)) .or. (.not. (y <= 1d+16))) then
        tmp = y * (y * -x)
    else
        tmp = x * (y - (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7.8e+43) || !(y <= 1e+16)) {
		tmp = y * (y * -x);
	} else {
		tmp = x * (y - (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7.8e+43) or not (y <= 1e+16):
		tmp = y * (y * -x)
	else:
		tmp = x * (y - (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7.8e+43) || !(y <= 1e+16))
		tmp = Float64(y * Float64(y * Float64(-x)));
	else
		tmp = Float64(x * Float64(y - Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7.8e+43) || ~((y <= 1e+16)))
		tmp = y * (y * -x);
	else
		tmp = x * (y - (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7.8e+43], N[Not[LessEqual[y, 1e+16]], $MachinePrecision]], N[(y * N[(y * (-x)), $MachinePrecision]), $MachinePrecision], N[(x * N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.8000000000000001e43 or 1e16 < y

   1. Initial program 99.8%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Step-by-step derivation

      1. distribute-lft-out-- 86.0%

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 1 - \left(x \cdot y\right) \cdot y} \]

      2. *-rgt-identity 86.0%

         \[\leadsto \color{blue}{x \cdot y} - \left(x \cdot y\right) \cdot y \]

      3. associate-*l* 71.2%

         \[\leadsto x \cdot y - \color{blue}{x \cdot \left(y \cdot y\right)} \]

      4. distribute-lft-out-- 85.0%

         \[\leadsto \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

   3. Simplified 85.0%

      \[\leadsto \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

   4. Taylor expanded in y around inf 85.0%

      \[\leadsto \color{blue}{-1 \cdot \left({y}^{2} \cdot x\right)} \]
   5. Step-by-step derivation

      1. unpow2 85.0%

         \[\leadsto -1 \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot x\right) \]

      2. associate-*r* 85.0%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot y\right)\right) \cdot x} \]

      3. mul-1-neg 85.0%

         \[\leadsto \color{blue}{\left(-y \cdot y\right)} \cdot x \]

      4. distribute-rgt-neg-out 85.0%

         \[\leadsto \color{blue}{\left(y \cdot \left(-y\right)\right)} \cdot x \]

      5. associate-*l* 99.8%

         \[\leadsto \color{blue}{y \cdot \left(\left(-y\right) \cdot x\right)} \]

      6. *-commutative 99.8%

         \[\leadsto y \cdot \color{blue}{\left(x \cdot \left(-y\right)\right)} \]

   6. Simplified 99.8%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot \left(-y\right)\right)} \]

   if -7.8000000000000001e43 < y < 1e16

   1. Initial program 99.9%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Step-by-step derivation

      1. distribute-lft-out-- 99.9%

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 1 - \left(x \cdot y\right) \cdot y} \]

      2. *-rgt-identity 99.9%

         \[\leadsto \color{blue}{x \cdot y} - \left(x \cdot y\right) \cdot y \]

      3. associate-*l* 99.9%

         \[\leadsto x \cdot y - \color{blue}{x \cdot \left(y \cdot y\right)} \]

      4. distribute-lft-out-- 99.9%

         \[\leadsto \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(y - y \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+43} \lor \neg \left(y \leq 10^{+16}\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \end{array} \]

Alternative 3: 91.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (* x (* y (- y))) (* y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = x * (y * -y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = x * (y * -y)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = x * (y * -y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = x * (y * -y)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(x * Float64(y * Float64(-y)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = x * (y * -y);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(x * N[(y * (-y)), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 99.8%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Step-by-step derivation

      1. distribute-lft-out-- 88.4%

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 1 - \left(x \cdot y\right) \cdot y} \]

      2. *-rgt-identity 88.4%

         \[\leadsto \color{blue}{x \cdot y} - \left(x \cdot y\right) \cdot y \]

      3. associate-*l* 76.2%

         \[\leadsto x \cdot y - \color{blue}{x \cdot \left(y \cdot y\right)} \]

      4. distribute-lft-out-- 87.6%

         \[\leadsto \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

   3. Simplified 87.6%

      \[\leadsto \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

   4. Taylor expanded in y around inf 84.1%

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot {y}^{2}\right)} \]
   5. Step-by-step derivation

      1. unpow2 84.1%

         \[\leadsto x \cdot \left(-1 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

      2. mul-1-neg 84.1%

         \[\leadsto x \cdot \color{blue}{\left(-y \cdot y\right)} \]

      3. distribute-rgt-neg-out 84.1%

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-y\right)\right)} \]

   6. Simplified 84.1%

      \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-y\right)\right)} \]

   if -1 < y < 1

   1. Initial program 100.0%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Step-by-step derivation

      1. distribute-lft-out-- 100.0%

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 1 - \left(x \cdot y\right) \cdot y} \]

      2. *-rgt-identity 100.0%

         \[\leadsto \color{blue}{x \cdot y} - \left(x \cdot y\right) \cdot y \]

      3. associate-*l* 100.0%

         \[\leadsto x \cdot y - \color{blue}{x \cdot \left(y \cdot y\right)} \]

      4. distribute-lft-out-- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

   4. Taylor expanded in y around 0 97.8%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (* y (* y (- x))) (* y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = y * (y * -x);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * (y * -x)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = y * (y * -x);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = y * (y * -x)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(y * Float64(y * Float64(-x)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = y * (y * -x);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(y * (-x)), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 99.8%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Step-by-step derivation

      1. distribute-lft-out-- 88.4%

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 1 - \left(x \cdot y\right) \cdot y} \]

      2. *-rgt-identity 88.4%

         \[\leadsto \color{blue}{x \cdot y} - \left(x \cdot y\right) \cdot y \]

      3. associate-*l* 76.2%

         \[\leadsto x \cdot y - \color{blue}{x \cdot \left(y \cdot y\right)} \]

      4. distribute-lft-out-- 87.6%

         \[\leadsto \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

   3. Simplified 87.6%

      \[\leadsto \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

   4. Taylor expanded in y around inf 84.1%

      \[\leadsto \color{blue}{-1 \cdot \left({y}^{2} \cdot x\right)} \]
   5. Step-by-step derivation

      1. unpow2 84.1%

         \[\leadsto -1 \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot x\right) \]

      2. associate-*r* 84.1%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot y\right)\right) \cdot x} \]

      3. mul-1-neg 84.1%

         \[\leadsto \color{blue}{\left(-y \cdot y\right)} \cdot x \]

      4. distribute-rgt-neg-out 84.1%

         \[\leadsto \color{blue}{\left(y \cdot \left(-y\right)\right)} \cdot x \]

      5. associate-*l* 96.3%

         \[\leadsto \color{blue}{y \cdot \left(\left(-y\right) \cdot x\right)} \]

      6. *-commutative 96.3%

         \[\leadsto y \cdot \color{blue}{\left(x \cdot \left(-y\right)\right)} \]

   6. Simplified 96.3%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot \left(-y\right)\right)} \]

   if -1 < y < 1

   1. Initial program 100.0%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Step-by-step derivation

      1. distribute-lft-out-- 100.0%

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 1 - \left(x \cdot y\right) \cdot y} \]

      2. *-rgt-identity 100.0%

         \[\leadsto \color{blue}{x \cdot y} - \left(x \cdot y\right) \cdot y \]

      3. associate-*l* 100.0%

         \[\leadsto x \cdot y - \color{blue}{x \cdot \left(y \cdot y\right)} \]

      4. distribute-lft-out-- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

   4. Taylor expanded in y around 0 97.8%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5: 57.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ y \cdot x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* y x))

C

double code(double x, double y) {
	return y * x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y * x
end function

Java

public static double code(double x, double y) {
	return y * x;
}

Python

def code(x, y):
	return y * x

Julia

function code(x, y)
	return Float64(y * x)
end

MATLAB

function tmp = code(x, y)
	tmp = y * x;
end

Wolfram

code[x_, y_] := N[(y * x), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
2. Step-by-step derivation

   1. distribute-lft-out-- 94.8%

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 1 - \left(x \cdot y\right) \cdot y} \]

   2. *-rgt-identity 94.8%

      \[\leadsto \color{blue}{x \cdot y} - \left(x \cdot y\right) \cdot y \]

   3. associate-*l* 89.4%

      \[\leadsto x \cdot y - \color{blue}{x \cdot \left(y \cdot y\right)} \]

   4. distribute-lft-out-- 94.4%

      \[\leadsto \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

3. Simplified 94.4%

   \[\leadsto \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

4. Taylor expanded in y around 0 60.4%

   \[\leadsto \color{blue}{y \cdot x} \]

5. Final simplification 60.4%

   \[\leadsto y \cdot x \]

Alternative 6: 2.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

   \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
2. Step-by-step derivation

   1. distribute-lft-out-- 94.8%

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 1 - \left(x \cdot y\right) \cdot y} \]

   2. *-rgt-identity 94.8%

      \[\leadsto \color{blue}{x \cdot y} - \left(x \cdot y\right) \cdot y \]

   3. associate-*l* 89.4%

      \[\leadsto x \cdot y - \color{blue}{x \cdot \left(y \cdot y\right)} \]

   4. distribute-lft-out-- 94.4%

      \[\leadsto \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

3. Simplified 94.4%

   \[\leadsto \color{blue}{x \cdot \left(y - y \cdot y\right)} \]
4. Step-by-step derivation

   1. *-un-lft-identity 94.4%

      \[\leadsto x \cdot \left(\color{blue}{1 \cdot y} - y \cdot y\right) \]

   2. distribute-rgt-out-- 94.4%

      \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(1 - y\right)\right)} \]

   3. associate-*l* 99.9%

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \left(1 - y\right)} \]

   4. flip-- 94.5%

      \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{1 \cdot 1 - y \cdot y}{1 + y}} \]

   5. associate-*r/ 93.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot \left(1 \cdot 1 - y \cdot y\right)}{1 + y}} \]

   6. metadata-eval 93.0%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot \left(\color{blue}{1} - y \cdot y\right)}{1 + y} \]

   7. +-commutative 93.0%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot \left(1 - y \cdot y\right)}{\color{blue}{y + 1}} \]

5. Applied egg-rr 93.0%

   \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot \left(1 - y \cdot y\right)}{y + 1}} \]
6. Step-by-step derivation

   1. *-commutative 93.0%

      \[\leadsto \frac{\color{blue}{\left(1 - y \cdot y\right) \cdot \left(x \cdot y\right)}}{y + 1} \]

   2. associate-/l* 93.8%

      \[\leadsto \color{blue}{\frac{1 - y \cdot y}{\frac{y + 1}{x \cdot y}}} \]

   3. *-commutative 93.8%

      \[\leadsto \frac{1 - y \cdot y}{\frac{y + 1}{\color{blue}{y \cdot x}}} \]

7. Simplified 93.8%

   \[\leadsto \color{blue}{\frac{1 - y \cdot y}{\frac{y + 1}{y \cdot x}}} \]

8. Taylor expanded in y around inf 39.9%

   \[\leadsto \frac{1 - y \cdot y}{\color{blue}{\frac{1}{x}}} \]

9. Taylor expanded in y around 0 3.0%

   \[\leadsto \color{blue}{x} \]

10. Final simplification 3.0%

    \[\leadsto x \]

Reproduce

herbie shell --seed 2023199 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))